Vertices of simple modules of symmetric groups labelled by hook partitions

In this article we study the vertices of simple modules for the symmetric groups in prime characteristic $p$. In particular, we complete the classification of the vertices of simple $S_n$-modules labelled by hook partitions

Foulkes modules and decomposition numbers of the symmetric group

The decomposition matrix of a finite group in prime characteristic p records the multiplicities of its p-modular irreducible representations as composition factors of the reductions modulo p of its irreducible representations in characteristic zero. The main theorem of this paper gives a combinatorial description of certain columns of the decomposition matrices of symmetric groups in odd prime characteristic. The result applies to blocks of arbitrarily high weight. It is obtained by studying the p-local structure of certain twists of the permutation module given by the action of the symmetric group of even degree 2m on the collection of set partitions of a set of size 2m into m sets each of size two. In particular, the vertices of the indecomposable summands of all such modules are characterized; these summands form a new family of indecomposable p-permutation modules for the symmetric group. As a further corollary it is shown that for every natural number w there is a diagonal Cartan number in a block of the symmetric group of weight w equal to w+1